Problem: $\lim_{x\to \frac{\pi}{2}}\tan(x)=?$ Choose 1 answer: Choose 1 answer: (Choice A) A $0$ (Choice B) B $\dfrac{1}{2}$ (Choice C) C $1$ (Choice D) D The limit doesn't exist.
Explanation: $\tan(x)$ is continuous on all points in its domain. Therefore, if $x=\dfrac{\pi}{2}$ is within the domain of $\tan(x)$, we can find $\lim_{x\to \frac{\pi}{2}}\tan(x)$ by direct substitution. $x=\dfrac{\pi}{2}$ is not in the domain of $\tan(x)$ : $\begin{aligned} \tan\left(\dfrac{\pi}{2}\right)&=\dfrac{\sin\left(\dfrac{\pi}{2}\right)}{\cos\left(\dfrac{\pi}{2}\right)} \\\\ &=\dfrac{1}{0} \end{aligned}$ Since direct substitution ends with $\dfrac{1}{0}$, we know that $\lim_{x\to \frac{\pi}{2}}\tan(x)$ doesn't exist.